mathematics, the Kronecker delta (named after
Leopold Kronecker) is a
function of two
variables, usually just positive
integers. The function is 1 if the variables are equal, and 0 otherwise:
where the Kronecker delta δij is a
piecewise function of variables i and j. For example, δ1 2 = 0, whereas δ3 3 = 1.
The Kronecker delta appears naturally in many areas of mathematics, physics and engineering, as a means of compactly expressing its definition above.
linear algebra, the n × n
identity matrix I has entries equal to the Kronecker delta:
where i and j take the values 1, 2, ..., n, and the
inner product of
vectors can be written as
The restriction to positive integers is common, but there is no reason it cannot have
negative integers as well as positive, or any discrete
rational numbers. If i and j above take rational values, then for example
This latter case is for convenience.